

Why is CRC is popular:




x^{3} + 2x + 1 ===> x^{3} + 1 (mod 2) (1 is odd and is mapped to 1) 
(x^{3} + x) + (x + 1) = x^{3} + 2x + 1 = x^{3} + 1 (mod 2) 
(x^{2} + x) × (x + 1) = (x^{3} + x^{2}) + (x^{2} + x) = x^{3} + 2x^{2} + x = x^{3} + x (mod 2) 
x^{2} + 1  x + 1 / x^{3} + x^{2} + x x^{3} + x^{2}  x x + 1  1 x^{3} + x^{2} + x  = x^{2} + 1 remainder 1 (mod 2) x + 1 
0 1 0 1 + 0 + 0 + 1 + 1     0 1 1 0 0 1 0 1  0  0  1  1     0 1 1 0 
Observation:

Bit string: 110101 ^^^^^^  + x^{0} + x^{1} + x^{2} + x^{3} + x^{4} + x^{5} Polynomial: x^{5} + x^{4} + x^{2} + 1 

(x^{3} + x) + (x + 1) = x^{3} + 2x + 1 = x^{3} + 1 (mod 2) 
(x^{2} + x) × (x + 1) = (x^{3} + x^{2}) + (x^{2} + x) = x^{3} + 2x^{2} + x = x^{3} + x (mod 2) 
x^{2} + 1  x + 1 / x^{3} + x^{2} + x x^{3} + x^{2}  x x + 1  1 x^{3} + x^{2} + x  = x^{2} + 1 remainder 1 (mod 2) x + 1 